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Mirrors > Home > MPE Home > Th. List > dftpos2 | Structured version Visualization version GIF version |
Description: Alternate definition of tpos when 𝐹 has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
dftpos2 | ⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmtpos 8248 | . . 3 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹) | |
2 | 1 | reseq2d 5987 | . 2 ⊢ (Rel dom 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = (tpos 𝐹 ↾ ◡dom 𝐹)) |
3 | reltpos 8241 | . . 3 ⊢ Rel tpos 𝐹 | |
4 | resdm 6033 | . . 3 ⊢ (Rel tpos 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = tpos 𝐹) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ (tpos 𝐹 ↾ dom tpos 𝐹) = tpos 𝐹 |
6 | df-tpos 8236 | . . . 4 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
7 | 6 | reseq1i 5983 | . . 3 ⊢ (tpos 𝐹 ↾ ◡dom 𝐹) = ((𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ↾ ◡dom 𝐹) |
8 | resco 6257 | . . 3 ⊢ ((𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ↾ ◡dom 𝐹) = (𝐹 ∘ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡dom 𝐹)) | |
9 | ssun1 4172 | . . . . 5 ⊢ ◡dom 𝐹 ⊆ (◡dom 𝐹 ∪ {∅}) | |
10 | resmpt 6044 | . . . . 5 ⊢ (◡dom 𝐹 ⊆ (◡dom 𝐹 ∪ {∅}) → ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡dom 𝐹) = (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡dom 𝐹) = (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) |
12 | 11 | coeq2i 5865 | . . 3 ⊢ (𝐹 ∘ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡dom 𝐹)) = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) |
13 | 7, 8, 12 | 3eqtri 2759 | . 2 ⊢ (tpos 𝐹 ↾ ◡dom 𝐹) = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) |
14 | 2, 5, 13 | 3eqtr3g 2790 | 1 ⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∪ cun 3945 ⊆ wss 3947 ∅c0 4324 {csn 4630 ∪ cuni 4910 ↦ cmpt 5233 ◡ccnv 5679 dom cdm 5680 ↾ cres 5682 ∘ ccom 5684 Rel wrel 5685 tpos ctpos 8235 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-fv 6559 df-tpos 8236 |
This theorem is referenced by: tposf12 8261 |
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